find the average value of the function below on the interval that is given. f(x) = 7x + 6; 4, 5

Explanation:

Step1: Recall average - value formula

The average value of a function $y = f(x)$ on the interval $[a,b]$ is given by $\bar{y}=\frac{1}{b - a}\int_{a}^{b}f(x)dx$. Here, $a = 4$, $b = 5$, and $f(x)=7x + 6$.

Step2: Calculate the integral

First, find $\int(7x + 6)dx=\frac{7}{2}x^{2}+6x+C$. Then, $\int_{4}^{5}(7x + 6)dx=\left[\frac{7}{2}x^{2}+6x\right]_{4}^{5}$. [ \begin{align*} \left(\frac{7}{2}(5)^{2}+6(5)\right)-\left(\frac{7}{2}(4)^{2}+6(4)\right)&=\left(\frac{7}{2}\times25 + 30\right)-\left(\frac{7}{2}\times16+24\right)\ &=\left(\frac{175}{2}+30\right)-\left(56 + 24\right)\ &=\left(\frac{175+60}{2}\right)-80\ &=\frac{235}{2}-80\ &=\frac{235 - 160}{2}\ &=\frac{75}{2} \end{align*} ]

Step3: Calculate the average value

Since $b - a=5 - 4 = 1$, the average value $\bar{y}=\frac{1}{5 - 4}\times\frac{75}{2}=\frac{75}{2}=37.5$.

Answer:

$37.5$